Figure, significant

9 State the number of significant figures in a given quantity. 

It is important in scientific work to make accurate measurements and to record them correctly. The recorded measurement should indicate the size of its uncertainty. One way to do this is to attach a to the recorded number. For example, if a bathroom scale indicates a person s weight correctly to within one pound, and a man reads the scale at 174 pounds, we would record his weight as 174 1 pound. The last digit, 4, is the uncertain digit. 

Another way to indicate uncertainty is to use significant figures. The number of significant figures in a quantity is the number of digits that are known accurately 

Begin with the first nonzero digit and end with the uncertain digit—the last digit shown. 

The uncertain digit is also called the doubtful digit. 

The first thing we need to learn is how to determine how many significant figures are in a number. 

Exact numbers. Some numbers are exact and have an infinite number of significant figures. Exact numbers occur in simple counting operations when you count 25 dollars, you have exactly 25 dollars. Defined numbers, such as 12 inches in 1 foot, 60 minutes in 1 hour, and 100 centimeters in 1 meter, are also considered to be exact numbers. Exact numbers have no uncertainty. 

205 has three significant figures 2.05 has three significant figures 61.09 has four significant figures 

Rules for significant figures should be memorized for use throughout the text 

One way of indicating that these zeros are significant is to write the number using scientific notation. Thus if the value 1000 has been determined to four significant figures, it is written as 1.000 X 10. If 590 has only two significant figures, it is written as 5.9 X 10.  

The way a result is written should tell something about the precision of the result. The more figures quoted the greater the implied precision. The significant figures are those that impart useful information. It would not be at all appropriate to quote the length of a swimming pool to a fraction of a millimeter. 

If it is not obvious then the number of significant figures is best determined by writing the result in scientific notation (i.e., x.xxx x 10y) and counting the digits. 

State how many significant figures there are in the following 

Expressing the concentration of copper in scientific notation, 0.00000572 becomes 5.72 x 10 6 M. Hence there are three significant figures, the digit before the decimal point and the two digits after the decimal point. 

20 tonnes of ammonium nitrate is equivalent to 5200 kg. Expressed in scientific notation the mass is 5.20 x 103kg and there are three significant figures. 

Quantities are composed of two parts a number (e.g., 5.31) and a unit (e.g., grams). The number must have the proper amount of significant figures, defined as follows  

Significant figures - the number of digits from the first nonzero digit on the left to the last nonzero digit on the right. 

The amount of significant figures in a number is perhaps determined easiest from the scientific notation for the number. Some examples are listed in Table C.l. 

The numbers 1,000 and 1,000. each have only 1 significant figure if one applies the definition above. This is wrong 1,000 has 1 significant figure and 1,000. has 4. We need to augment the definition as follows  

One operational way to identify significant zeros is to convert the number into scientific notation. If the decimal point does not cross a zero when converting a number into scientific notation, the zero is always significant. Some additional examples are given below (significant figures are in bold). 

Numbers that represent discrete objects have no uncertainty. If five measurements were made, and we calculate the average by adding the five results and dividing by 5, the number 5 has no effect on the number of significant figures in the answer. There is no uncertainty in the number of measurements made, so the 5 can be considered to have an infinite number of significant figures. 

The following are some rules that should be observed when reporting results  

In enumerating data, report all significant figures, such that only the last figure is uncertain. 

If a calculation involves a combination of mnltiplication (or division) and addition (or subtraction), the steps must be treated separately. When nsing a calculator or spreadsheet program, the best approach is to keep all significant figures throughout the calculation and round off at the end. 

In performing pharmacokinetic calculations, we must take care to get the most precise answer that can be supported by the data we have. Conversely, we do not want to express our answer with greater precision than we are justified in claiming. The rules of significant figures will help us with this task. These are listed in Box 2.3. 

4 shows some typical units used in pharmacokinetics as well as the mathematical rules which apply to units. 

Throughout the text, various equivalent units will be mentioned at intervals so that the student will become adept at recognizing them. For example, 1.23 pgmL can also be expressed as 1.23mcgmL or 1.23mgL . Micrograms can be expressed as pg or meg (not an S.I. unit but commonly used to avoid any confusion 

Authors have a tendency to report numbers that do not accurately reflect the actual performance capability of the method. Modern data systems will provide multiple numbers after the decimal place, but it is the responsibility of the analyst to determine the appropriate number of significant figures that should be reported. As a general approach, the method precision should inform the reporting of results. The precision indicates the number at which uncertainty occurs, and this should be the final digit reported. For example, if the standard deviation for a method is determined to be 0.1, it is not appropriate to report a result as 1.463—the result should be reported as 

as adding additional numbers after the decimal gives a false impression of the measurement capabilities of the method. However, it is important that results never be rounded up so that a regulatory or other limit is breached, as this could be challenged in a court of law. 

Suppose that you measure the density of a mineral by finding its mass (4.635 0.002 g) and its volume (1.13 0.05 mL). Density is mass per unit volume 4.635 g/1.13 mL = 4.101 8 g/mL. The uncertainties in mass and volume are 0.002 g and 0.05 mL, but what is the uncertainty in the computed density And how many significant figures should be used for the density This chapter answers these questions and introduces spreadsheets—a powerful tool that will be invaluable to you in and out of this course. 

Significant figures minimum number of digits required to express a value in scientific notation without loss of precision 

You should write one of these three numbers, instead of 92 500, to indicate how many figures are actually known. 

Zeros are significant when they are (1) in the middle of a number or (2) at the end of a number on the right-hand side of a decimal point. 

Interpolation estimate all readings to the nearest tenth of the distance between scale divisions 

The numerical value of every observed measurement is an approximation. No physical measurement, such as mass, length, time, volume, velocity, is ever absolutely correct. The accuracy (reliability) of every measurement is limited by the reliability of the measuring instrument, which is never absolutely reliable. 

Consider that the length of an object is recorded as 15.7 cm. By convention, this means that the length was measured to the nearest tenth of a centimeter and that its exact value lies between 15.65 and 15.75 cm. If this measurement were exact to the nearest hundredth of a centimeter, it would have been recorded as 

The value 15.7 cm represents three significant figures (1, 5, 7), while 15.70 cm represents four significant figures (1, 5, 7, 0). A significant figure is one which is known to be reasonably reliable. 

In elementary measurements in chemistry and physics, the last digit is an estimated figure and is considered as a significant figure. 

Some numerical values are exact to as many significant figures as necessary, by definition. Included in this category are the numerical equivalents of prefixes used in unit definition. For example, 1 cm = 0.01 m by definition, and the units conversion factor, 1.0 x 10 m/cm, is exact to an infinite number of significant figures. 

Addition and subtraction the answer should have the same number of decimal places as the quantity with the fewest number of decimal places. 

Multiplication and division the answer should not contain a greater number of significant figures than the number in the least precise measurement. 

Rounding when rounding, examine the number to the right of the number that is to be the last. Round up if this number is 5, round down if 5. If the last number is equal to 5 then round up if the resulting number is even, if odd then round down. For example, 5.678 becomes 5.68 to 3 significant figures, but 5.673 becomes 5.67. 5.675 becomes 5.68, but 5.665 becomes 5.66. 

This discussion applies to numerical values of variables whose possible range constitutes a continuum in computer terminology these are called real numbers, as opposed to integers. Integers should be identified as such, and all of the digits necessary to express them should be given. 

The use of too few digits in reporting numerical values robs the user of valuable information. The usual fault, however, is the reckless use of too many digits, which conveys a false sense of the accuracy or precision of the numerical value. 

In reporting numerical values, certain rules regarding significant figures should be followed. 

Numerical values are considered to be uncertain in the last digit by 3 or more, and perhaps slightly uncertain in the next-to-last digit. Ordinarily the next-to-last digit should not be uncertain by more than 2. 

In rounding off numbers, (a) increase the last retained digit by 1 if the leftmost digit to be dropped is more than 5, or is 5 followed by any nonzero digits  

The certain digits and the first uncertain digit of a measurement 

We have seen that any measurement involves an estimate and thus is uncertain to some extent. We signify the degree of certainty for a particular measurement by the number of significant figures we record. 

Because doing chemistry requires many types of calculations, we must consider what happens when we use numbers that contain uncertainties. 

Exact numbers. Often calculations involve numbers that were not obtained using measuring devices but were determined by counting  

10 experiments, 3 apples, 8 molecules. Such numbers are called exact numbers. They can be assumed to have an unlimited number of significant figures. Exact numbers can also arise from definitions. For example, 

Since the pointer is between the 147- and 148-lb markings, mentally divide the space between the markings into 10 equal spaces and estimate the next digit. In this case, the result should be reported as  

What if you estimated a little differently and wrote 147.6 lb In general, one unit of difference in the last digit is acceptable because the last digit is estimated and different people might estimate it slightly differently. However, if you wrote 147.2 lb, you would clearly be wrong. 

Interior zeros (zeros between two numbers) are significant. 

Trailing zeros (zeros to the right of a nonzero number) that fall after a decimal point are significant. 

FIGURE 2.4 Additional examples of nondigital measurements, in this case classical thermometers, (a) 22.17° and (b) 32.50°. 

Measuring a pin. (a) The length is between 2.8 cm and 2.9 cm. (b) Imagine that the distance between 2.8 and 2.9 is divided into 10 equal parts. The end of the pin occurs after about 5 of these divisions. 

Note that the first two digits in each measurement are the same regardless of who made the measurement these are called the certain numbers of the measurement. However, the third digit is estimated and can vary it is called an uncertain number. When one is making a measurement, the custom is to record all of the certain numbers plus the first uncertain number. It would not make any sense to try to measure the pin to the third decimal place (thousandths of a centimeter), because this ruler requires an estimate of even the second decimal place (hundredths of a centimeter). 

This example illustrates the method of unit balancing. By this means, we have multiplied and divided the units as if they were numbers combining and simplifying the products and quotients to reveal the answer is in the desired units, ft /s. Dimensional analysis is an extremely useful engineering tool. It serves as a check on the appropriateness of the units that were employed in the calculations. Equally important, it is a check on the validity of the problemsolving methodology and the correctness of the mathematical operations. 

For example, suppose that we add the following numbers 3.51, 2.205, and 0.0142. The sum is 5.7292, but it should be reported to three significant figures as 5.72, or rounded to 5.73. Since one of the numbers used in the sum was accurate only to the nearest hundredth, it would be presumptuous and misleading to report the answer to the thousandth or ten-thousandth place. 

In multiplication and division, the following rule should be used to determine the number of significant figures in the answer The product or quotient should contain the number of significant digits that are contained in the number with the fewest significant digits. For example, the product 

If the number 75.22 is divided by 25.1, the quotient should be reported as 3.00, not 2.9968, since the last two digits are unreliable and not significant. 

If the number following the least significant figure is equal to 5, it is not clear whether the number should be rounded up or down. For such instances, many engineers use the following practice Truncate the number or round it up so that the recorded number is even. With this rule, the number 2.525 would be truncated to three significant figures as 2.52 the number 2.535 would be rounded up to 2.54. 

Can a set of measurements be precise without being accurate Can the average of a set of measurements be accurate and the individual measurements be 

Consider these measurements made on a low-precision balance 10.4,10.2, and 10.3 g. The reported result is best expressed as their average, that is, 10.3 g. 

The quantity shown here, 0.004004500, has seven significant figures. All nonzero digits are significant, as are the indicated zeros. 

Not significant zeros used only to locate the decimal point 

Significant zeros at the end of a number to the right of decimal point 

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The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

On entering SigmaPlot (we use version 5.0), one is presented with a data table that is essentially a spreadsheet. Enter T as the independent or A -variable into the first eolumn of the SigmaPlot data table and Cp/T as the dependent or y-variable into the second column. The SigmaPlot data table should resemble columns 1 and 3 of Table 1-3. Rounding to three significant figures is permissible. 

Using MOPAC and the MNDO Hamiltonian, calculate the energy and equilibrium bond length of N2 to 4 significant figures. The input file is... 

Relations which are exact are indicated by an asterisk ( ). Factors in parentheses are also exact. Other factors are within 5 in the last significant figure. 

Formula weights are based upon the International Atomic Weights of 1993 and are computed to the nearest hundredth when justified. The actual significant figures are given in the atomic weights of the individual elements. 

Exact numbers, such as the stoichiometric coefficients in a chemical formula or reaction, and unit conversion factors, have an infinite number of significant figures. A mole of CaCb, for example, contains exactly two moles of chloride and one mole of calcium. In the equality... 

Significant figures are also important because they guide us in reporting the result of an analysis. When using a measurement in a calculation, the result of that calculation can never be more certain than that measurement s uncertainty. Simply put, the result of an analysis can never be more certain than the least certain measurement included in the analysis. 

Substituting known volumes (with significant figures appropriate for pipets and volumetric flasks) into equation 2.4... 

There are a few basic numerical and experimental tools with which you must be familiar. Fundamental measurements in analytical chemistry, such as mass and volume, use base SI units, such as the kilogram (kg) and the liter (L). Other units, such as power, are defined in terms of these base units. When reporting measurements, we must be careful to include only those digits that are significant and to maintain the uncertainty implied by these significant figures when transforming measurements into results. 

Round each of the following to three significant figures,... 

Report results for the following calculations to the correct number of significant figures. 

Calculate the molar concentration of NaCl, to the correct number of significant figures, if 1.917 g of NaCl is placed in a beaker and dissolved in 50 mF of water measured with a graduated cylinder. This solution is quantitatively transferred to a 250-mF volumetric flask and diluted to volume. Calculate the concentration of this second solution to the correct number of significant figures. 

The numerator, therefore, is 23.41 0.028 (note that we retain an extra significant figure since we will use this uncertainty in further calculations). To complete the calculation, we estimate the relative uncertainty in Ca using equation 4.7, giving... 

In this experiment students measure the length of a pestle using a wooden meter stick, a stainless-steel ruler, and a vernier caliper. The data collected in this experiment provide an opportunity to discuss significant figures and sources of error. Statistical analysis includes the Q-test, f-test, and F-test. 

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